Note that,
\(\mathcal {F}^M_t \subseteq \mathcal {F}^{M,S}_t\)  and
that
\(\mathcal {F}^S_t \subseteq \mathcal {F}^{M,S}_t\) .
Consequently, every
\(\mathcal {F}^{M,S}_t\) -martingale is a \(\mathcal {F}^M_t\) -martingale and, at the same time, every
\(\mathcal {F}^{M,S}_t\) -martingale is a \(\mathcal {F}^S_t\) -martingale.
But, in general,
it is not true that a \(\mathcal {F}^M_t\) -martingale is a
\(\mathcal {F}^{M,S}_t\) -martingale
and that a
\(\mathcal {F}^S_t\) -martingale is a
\(\mathcal {F}^{M,S}_t\) -martingale.
This is a classical topic in this context, the so-called problem of the enlargement of filtrations.
An exciting example is Azema's martingale, see Subsection 4.3.8 in Jeanblanc et al.
[1]}.
Hence, to overcome this difficulty,
we assume
a property for the class of martingales, that is the Immersion property as given below.
